Let $(X,\tau)$ be a topological space and let $f_1,\ldots,f_n:X\to\mathbb{R}$ be continuous functions (the topology of $\mathbb{R}$ is the usual one). Define $g:X\to\mathbb{R}$ by $g(x)=\text{max}\{f_1(x),\ldots,f_n(x)\}$. How to show that $g$ is continuous?
I showed that, if $I\subset \mathbb{R}$ is open, and $V_i$ is the pre-image of $I$ by $f_i$ (which is open in $X$), then $g^{-1}(\{I\})\subset \bigcup_{i=1}^n V_i$. But this does not say much.
Best Answer
You can first prove the statement for two functions and then proceed by induction. To show that $\max(f,g):X \to \mathbb R$ is continuous note that $\max(x,y) = \frac{x+y} 2 + \frac{\mid x-y \mid } 2$, for all $x,y \in \mathbb R$. Now, you only need to show that $\mid f \mid$ is continous, if $f$ is. But this is a composition of two continuous functions, so we are done.